Geometry and Gâteaux smoothness in separable Banach spaces

[EN] It is a classical fact, due to Day, that every separable Banach space admits an equivalent Gateaux smooth renorming. In fact, it admits an equivalent uniformly Gateaux smooth norm, as was shown later by Day, James, Swaminathan, and independently by the third named author. It is therefore rather unexpected that the existence of Gateaux smooth renormings satisfying various quantitative estimates on the directional derivative has rather strong structural and geometrical implications for the space. For example, by a result of Vanderwerff, if the directional derivatives satisfy a p-estimate, where p varies arbitrarily with respect to the point and the direction in question, then the Banach space must be an Asplund space. In the present survey paper, we discuss the interplay between various types of Gateaux differentiability of norms and extreme points with the geometry of separable Banach spaces. In particular, we present various characterizations of Asplund, reflexive, superreflexive, and other classes of separable Banach spaces, via smooth as well as rotund renormings. We also include open problems of various levels of difficulty, which may foster research in the area of smoothness and renormings of Banach spaces.Supported in part by GACR P201/11/0345. Supported in part by Proyecto MTM2008-03211. Ministerio de Ciencia e Innovaci ´on, by a grant BEST 2010-134 of the Generalitat Valenciana, and by a grant from the Universidad Polit´ecnica de Valencia, PAID 2009, Spain. Supported in part by a grant AVOZ 101 905 03 and IAA 100190901 (Czech Republic).Hajek, P.; Montesinos Santalucia, V.; Zizler, V. (2012). Geometry and Gâteaux smoothness in separable Banach spaces. Operators and Matrices. 6(2):201-232. https://doi.org/10.7153/oam-06-152012326